False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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An extension of nilpotent groups is nilpotent

Statement

False. If $N ⊴ G$ and both $N$ and $G / N$ are nilpotent, then $G$ is nilpotent.

Facts & Assumptions

Given: The normal subgroup $A_{3} ⊴ S_{3}$ .

[F1]

A nontrivial group has nilpotency class one exactly when it is abelian (Nilpotent groups and nilpotency class).

[F2]

A group is nilpotent exactly when it has a central series from $1$ to the whole group (Nilpotence via central series, the upper central series, and the lower central series).

[L2]

A surjective homomorphism induces an isomorphism from the quotient by its kernel onto its image (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

Refutation

technique · direct

1.1

The group $A_{3}$ is cyclic of order three, and the sign map has kernel $A_{3}$ and image $C_{2}$ , so [L2] gives $S_{3} / A_{3} ≅ C_{2}$ .

givenL2algebra

1.2

The center of $S_{3}$ is trivial: no nonidentity permutation commutes with both $(12)$ and $(123)$ . If a nontrivial group had a central series, its first nontrivial term would lie in its center; hence [F2] shows that $S_{3}$ is not nilpotent.

F2algebra

2.1

Both $A_{3}$ and $S_{3} / A_{3}$ are nontrivial abelian groups and hence nilpotent by [F1].

step 1.1F1

3.1

Therefore $1 \to A_{3} \to S_{3} \to C_{2} \to 1$ is an extension of nilpotent groups whose middle group is not nilpotent.

step 1.1step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
K. Conrad, Subgroup Series I (standard reference, not scraped)
K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)